"Mathematics is permanent revolution." That's one of my favorite sentences from Bob and Ellen Kaplan's The Art of the Infinite. It works on so many levels: on the one hand they're saying that the great revolutionary ideas of mathematics are continually being born out of and built upon the ideas that preceded them. They are, on the other hand, also making a pun about going around in circles. And they are, on yet another hand, deliberately recalling the structure of the book itself: the ideas always seem to be progressing forward, but every once in a while the authors force us to take our noses out of the details and appreciate that the idea we're exploring is really something we have already explored in an earlier chapter, we're just seeing it from a higher vantage point or a different angle. We've gone in a big circle and come back. And they're saying that this reflects the structure of mathematics itself: Mathematics is an ever-expanding spiral. The frontispiece of the book is "The Tower of Mathematics." It is an inverted Tower of Babel: as we go up it gets more coherent and the physical structure gets wider to accommodate the greater scope of the ideas. The paragraph that begins with this loaded sentence ends by quoting William Blake on opening the doors of our perception which of course makes you think of Jim Morrison and political revolution. And you realize they are also deliberately evoking Marx and Engels who first called for permanent revolution, which Trotsky described as "a revolution whose every successive stage is rooted in the preceding one," which is pretty much how mathematical revolutions proceed. This would be the perfect place to end this paragraph but I can't resist telling you the end of the Trotsky quote, "The permanent revolution means ... a revolution whose every successive stage is rooted in the preceding one and which can end only in complete liquidation." I guess I'm hoping this last part doesn't apply to mathematics. (And, speaking of the need for revolution, I'm also hoping that John Ashcroft isn't tracking visitors to the Leon Trotsky Internet Archive, where I found this quote.)

Nearly every paragraph of prose in the book is similarly layered and textured, jam-packed with literary, philosophical and historical allusion. The Kaplans write like humanities professors. You could spend months picking it all apart and still you'd probably miss most of it. I read a lot of mathematical exposition and we all seem to communicate in the same way — we use a simple spare prose style. We strive for complete clarity and unambiguity. Martin Gardner's clear, simple prose is our ideal. We're mathematicians, that's how mathematics is, so of course it's how we talk. If you've ever been to a faculty meeting, you know that our colleagues in the humanities communicate differently. For them it is a mark of erudition to slather a sentence with levels of meaning and subtle allusion. A word for a mathematician ideally conveys one single distinct concept; for your humanist a word weighs a ton — he won't let a word get away with doing only one job, if he can squeeze three out of it. I suspect this has something to do with your average humanist's disdain for the notion that mathematics is a creative art. To him layers of meaning and indirection seem essential to artistic communication. Mathematicians know what will happen if we allow auditors to interpret our utterances on their own — everyone will interpret things in his own subtly different way and soon we won't be talking about the same things. For a humanist, the individual, personal, interaction with the ideas is the whole point. Perhaps we communicate differently because, at some fundamental level, our notions of what truth is and our purposes in communicating it are different.

Regular readers of FOCUS Online will not find very much mathematics in this book that they don't already know. For me, one exception is the proof, attributed by the Kaplans to James Clarkson, that the series of prime reciprocals diverges. There are no detours into infinite products or complex analysis. The proof is truly elementary, though not easy: the authors describe it as "wonderfully acrobatic," which is about right. I can't resist telling the story of my favorite "proof" of this fact. When I was in graduate school a bunch of us were sitting around and somehow this very question came up. None of us knew whether the series converged and none of us could make even a convincing plausibility argument either way. I went down the hall and asked Joe Christie, a newly arrived post-doc from Berkeley with an encyclopedic knowledge of mathematics and its literature. The guy could always give you chapter and verse on any question and tell you three different places to look it up and the strengths and weaknesses of the sources. When I asked him whether the series 1/p converged he looked up at the ceiling, thought for a second and said, "It must diverge, because if it converged, then the number to which it converged would be very famous, and I've never heard of it." Joe would have been familiar with most of the content of the Kaplans' book: they start with the counting numbers, then figurate numbers, rationals, Hippasus's proof of the irrationality of the square root of two (after which they comment, "You may now find yourself in the distracted state where mathematicians notoriously live. The genie you rubbed from its bottle was much more powerful than you thought: barely under control.") and they wind up the first chapter with a discussion of complex numbers. The other chapters are equally rich: axiomatics (including the Hilbert-Brouwer dispute) in chapter two, elementary number theory in chapter three, infinite series in chapter four, Euclidean geometry in chapter five, ruler and compass constructions in chapter six, eiπ in chapter seven, projective geometry in chapter eight, winding up with Cantorian set theory in the final chapter. This catalogue of topics does the book a disservice — the real purpose of the book is not to present a collection of cool math ideas; no, the purpose is to lay bare the soul of mathematics and in that purpose the Kaplans succeed like no one else writing about mathematics today.

The Kaplans run a school for mathematics in Boston called The Math Circle. (See http://themathcircle.org/.) Their philosophy is that everyone can learn mathematics and can learn to enjoy mathematics and, in fact, can create mathematics. They approach mathematics with a playful, daring spirit of exploration, which they call "the Alcibiades humor." This spirit is the true strength of the book. The reader is carried along on a voyage of discovery by a pair of guides whose wit and wisdom and delight in what they're doing leaps off of every page. As I read I marked potential passages to quote in this review, when I was done I found nearly a hundred such passages. I'll leave you with two that seemingly contradict one another, but like Whitman, the Kaplans are large, they contain multitudes:

The rationals, the reals, the complex numbers no longer appear as successive approximations to what ultimately is, but as ever more tenuous fictions flung in support around the central keep.

And, later, after a discussion of Dirichlet's theorem on primes in arithmetic progression and as prelude to the Prime Number theorem:

Once again we are at a loss in trying to see the structure of the primes: no particular rhythm carries more of them than another. Yet if we assume chaos we cannot but deduce despair. Since intuition and common sense have left us stranded, we need an insight — and then a proof for it to nestle comfortably into. Gauss — whom we saw as a schoolboy triumphantly writing on his slate — used to contemplate tables of primes for sheer amusement, the way Russians always and the English on country house weekends love browsing through railway timetables.

One unanticipated consequence of the recent upsurge in popular interest in mathematics has been the proliferation of new voices in mathematical exposition. I read a lot of math exposition, but even I can't keep up with everything that's appearing lately. The thing that most pleases me about this book is the appearance of, not just a new voice, but a completely new style: mathematics as humanistic exploration of the human condition. We know that mathematics is not a straightforward application of logic to axioms. It has a complicated, messy history and philosophy, it progresses in fits and starts with loops of feedback and chaotic bits we don't quite understand. The Kaplans rejoice in mathematics' humanity and you'll rejoice along with them.

A few weeks ago I was browsing the "Odd, Unusual, and Bizarre" table at my local Barnes and Noble and I discovered, in between A Field Guide to Stains and Famous Last Words, a copy of the Kaplans' first book, The Nothing That Is: A Natural History of Zero. Rescue their new book from that oblivion: Buy this book. And buy an extra copy for your brother who majored in history and doesn't really get what it is that you do and refuses to believe that mathematics is an art.